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The mathematical stance

Abstract

Defenders of the enhanced indispensability argument argue that the most effective route to platonism is via the explanatory role of mathematical posits in science. Various compelling cases of mathematical explanation in science have been proposed, but a satisfactory general philosophical account of such explanations is lacking. In this paper, I lay out the framework for such an account based on the notion of “the mathematical stance.” This is developed by analogy with Dennett’s well-known concept of “the intentional stance.” Roughly, adopting the mathematical stance towards a particular physical phenomenon involves treating it as an abstract mathematical structure for the purposes of prediction and explanation. Interestingly, Dennett himself frequently draws analogies between his intentional stance towards beliefs and desires and scientists’ stance towards centers of gravity. I explore the theoretical role played by centers of gravity within science and discuss how an indispensabilist platonist ought to categorize the ontological status of this type of posit. I conclude with some thoughts on how an approach based on the mathematical stance might be developed into a more general philosophical account of the application of mathematics in science.

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Notes

  1. Bueno & Colyvan [2011, p. 347].

  2. ibid.

  3. For further elaboration of this problem see Nguyen & Frigg [2017]. Their proposed solution involves the notion of a “structure generating description,” which uses physical properties to carve up a given system into objects and relations from which the resulting structure can be abstracted. However it is not clear that this solves the Assumed Structure Problem rather than simply pushing the question back to what makes one structure generating description the uniquely correct one.

  4. See Baker [2009].

  5. See e.g. Leng [2010, Chapter 6].

  6. In the sense that a line drawn vertically down from the center of gravity does not meet any part of the base.

  7. Baker [2017].

  8. Because any water added will immediately be above the height of the center of gravity of the glass.

  9. For details of the cicada example, see Baker [2005]. For the bridges of Konigsberg example, see Pincock [2007].

  10. This is not the only reason. One problem with the honeycomb example is that the mathematics concerns a 2-dimensional optimization problem whereas actual bees are facing a 3-dimensional optimization problem. Another problem is that there are alternative, non-optimization-based explanations for the hexagonal shape that are based just on the physical behavior of liquid wax as it forms into cells. For more details, see Raz [2013].

  11. Baker [2005, p. 228].

  12. Lange [2018]. Lange actually couches the example in terms of center of mass, not center of gravity, but the core point remains the same.

  13. Dennett [1992].

  14. For further discussion of how (and where) to draw the boundary between abstract and concrete, see Rosen [2017], especially the section on the non-spatiality criterion.

  15. One of the interesting (and complicating) factors here is that centers of gravity tend to play a role in scientific explanations that is more token-based than type-based. This is unlike more canonical mathematical explanations. Numbers may play an explanatory role for one physical phenomenon (e.g. cicada periods), and those very same numbers may also play an explanatory role in a quite distinct physical phenomenon (e.g. bicycle gear ratios). In the case of centers of COG’s, by contrast, each different stability explanation is likely to involve a distinct COG.

  16. Dennett [1988, p. 496].

  17. ibid.

  18. Dennett [2009, p. 349].

  19. Dennett [1991, p. 27].

  20. Dennett [2009, p. 340].

  21. Different predictive and explanatory goals may also yield different levels of structural stance. The physical stance is at a lower level than any structural stance (and I leave open whether the physical stance should itself be considered structural, since this hinges on more general issues concerning structural realism in the philosophy of science).

  22. Perhaps these two categories of case can be brought closer together by thinking of the first example as also (in a sense) involving the violation of an idealizing assumption. The assumption in this case is that by representing each bridge as a link in the graph, each link thereby represents a viable crossing.

  23. Dennett [2009, p. 341].

  24. ibid.

  25. Dennett [1988, p. 497].

  26. Dennett [1987, p. 72].

  27. Dennett [1992, p. 104]. As Richard Joyce puts it, “Dennett is adamant that his psychological instrumentalism allows for the real existence of beliefs and desires, but as “abstracta” rather than neurological events.” (Joyce [2013, p. 517])

  28. e.g. “Belief is a perfectly objective phenomenon.” (Dennett [1987, p. 15])

  29. Dummett [1978, p. xxxviii].

  30. See e.g. Shapiro [1997].

  31. Dennett [1987, p. 72].

  32. op. cit., pp. 72–3.

  33. Demeter [2009, p. 62].

  34. Thus while (for example) recent work on mathematical explanation within mathematics has drawn attention to the potential explanatory value of proofs, the target of the explanation is a result in pure mathematics, and thus it only has ‘mathematics-independent’ predictive or explanatory value when it is itself applied.

  35. “The “dramatic idiom” of intentional attribution must be taken seriously, but not too seriously, treating it always as what I call a “heuristic overlay” or a “stance.” (Dennett [1988, p. 505]) The term “heuristic overlay” traces back to Dennett [1969].

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Baker, A. The mathematical stance. Synthese 200, 53 (2022). https://doi.org/10.1007/s11229-022-03458-8

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